ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-11 Unicode version

Axiom ax-11 1438
Description: Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent  A. x ( x  =  y  ->  ph ) is a way of expressing " y substituted for  x in wff  ph " (cf. sb6 1809). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases.

Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1750, ax11v2 1743 and ax-11o 1746. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
ax-11  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )

Detailed syntax breakdown of Axiom ax-11
StepHypRef Expression
1 vx . . 3  setvar  x
2 vy . . 3  setvar  y
31, 2weq 1433 . 2  wff  x  =  y
4 wph . . . 4  wff  ph
54, 2wal 1283 . . 3  wff  A. y ph
63, 4wi 4 . . . 4  wff  ( x  =  y  ->  ph )
76, 1wal 1283 . . 3  wff  A. x
( x  =  y  ->  ph )
85, 7wi 4 . 2  wff  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) )
93, 8wi 4 1  wff  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff set class
This axiom is referenced by:  ax10o  1645  equs5a  1717  sbcof2  1733  ax11o  1745  ax11v  1750
  Copyright terms: Public domain W3C validator